Midterm Exam Problem 1. Let M be a smooth manifold. Suppose V and W are smooth n i ∂ nector fields on M and ω is a smooth one-form on M. Let V = Pi=1 V ∂xi and n i ∂ n i W = Pi=1 W ∂xi , and ω = Pi=1 ωidx be the coordinate expressions for V , W and ω in terms of some smooth local coordinates (xi) for M. For a smooth function f ∈ C∞(M), we define (*) LV f = Vf. (i) Show that LV W =[V,W] has the coordinate expression n ∂Wj ∂V j ∂ L W = X (V i − W i ) . V ∂xi ∂xi ∂xj i,j=1 (ii) The one-form LV ω is defined by the product rule LV (ω(W )) = (LV ω)(W )+ω(LV W ), for all smooth vector fields W, in which ω(W ) is simply a smooth function and LV (ω(W )) is defined by (*). Deduce that LV ω has the coordinate expression n ∂ω ∂V i L ω = X (V i j + ω )dxj . V ∂xi i ∂xj i,j=1 k (iii) Let D be a connection on M with the Christoffel symbol Γij . Define the 1-form DV W by the product rule DV (ω(W )) = (DV ω)(W )+ω(DV W ), for all smooth vector fields W. Show that the coordinate expression for DV W is n i i j k DV ω = X(V ∂iωk − V ωjΓik)dx . j=1 . Problem 2. Let M be a smooth manifold. (1) Show that the map L :Γ(TM) × Γ(TM) → Γ(TM), with (X, Y ) 7→ LX Y (the Lie derivative), is not a connection. (2) Show that there is a vector field on R2 that vanishes along the x1-axis, but whose ∂ 1 Lie derivative with respect ∂x1 does not vanish on the x -axis. Problem 3. Prove that the Lie algebra of any abelian Lie group is abelian. Typeset by AMS-TEX 1 2 Problem 4. Notice the following Definition. A vector field W is said to be invariant under a flow θ if (θt)∗Wp = Wθt(p) for all (t, p) in the domain of θ. Let V and W be smooth vector fields on M, with flows θ and ψ, respectively. Show that the following are equivalent: (1) [V,W]=0. (2) LV W =0. (3) LW V =0. (4) W is invariant under the flow of V . (5) V is invariant under the flow of W . (6) θt ◦ ψs = ψs ◦ θt whenever either side is defined. ⇒ (Hint: Show (2) (4) by considering X(t)=(θ−t)∗(Wθt(p)) and deriving that 0 d X (t0)=(θ−t )∗ (θ−s)∗Wθ (θ (p)) =(θ−t )∗(LV W ), making the change of 0 ds s t0 0 s=0 variables t = t0 + s.) Problem 5. Let V1, ··· ,Vk be smooth independent vector fields on an open subset O of a smooth manifold M. Choose a smooth chart (U, (xi)) centered at p. By rearranging the coordinates if necessary, we may assume that the vectors ∂ ∂ (V1 , ··· ,Vk , , ) p p ∂xk+1 ∂xn p p span TpM. Let θi denote the flow of Vi for i =1, ··· ,k. Then there exist ε>0 and a neighborhood W of p such that the composition (θk)tk ◦ (θk−1)tk−1 ◦···◦(θ1)t1 is defined on W and maps W into U whenever |t1|, ···|tk| <ε. Let S = {(uk+1, ··· ,un):(0, ··· , 0,uk+1, ··· ,un) ∈ W }, and define ψ :(−ε, ε)k × S → U by 1 k k+1 n k+1 n ψ(u , ··· ,u ,u , ··· ,u )=(θk)uk ◦···◦(θ1)u1 (0, ··· , 0,u , ··· ,u ). ≤ ≤ ∂ ··· (a) Suppose that [Vi,Vj ]=0for1 i, j k. Show that ψ∗ ∂ui = Vi, for i =1, ,k. (Hint: Use (6) in Problem 4). ∂ ∂ (b) Observe that ψ∗ i = i for i = k +1, ··· ,n. ∂u 0 ∂x p Use this, (1) and the inverse function theorem to show that ψ is a diffeomorphism in a neighborhood of 0 if [Vi,Vj ] = 0 for 1 ≤ i, j ≤ k. (c) Show that the following are equivalent: (1) [Vi,Vj ] ≡ 0, ∀1 ≤ i, j ≤ k,inO. (2) There exist smooth coordinates (u1, ··· ,un) in a neighborhood of each point O ≡ ∂ ··· of such that Vi ∂ui , i =1, ,k. 3 Problem 6. Let D0 and D1 be two connections on a smooth manifold M. 2 (a) Show that the difference between them defines a 1-tensor field A by 1 0 A(X, Y )=DX Y − DX Y, for X, Y ∈ Γ(TM), called the difference tensor. Thus, if D0 is any linear connection on M, the set of all connections is precisely 2 {D0 + A : A is a smooth -tensor}. 1 (b) Show that D0 and D1 determine the same geodesics iff their difference tensor is antisymmetric; i.e. A(X, Y )=−A(Y,X), for X, Y ∈ Γ(TM). (c) Show that D0 and D1 have the same torsion tensor iff their difference tensor are symmetric; i.e. A(X, Y )=A(Y,X), for X, Y ∈ Γ(TM). Problem 7. For any fixed R>0, the following Riemannian manifolds are all mutually isometric. n (1) Hyperbolic Model HR is the “upper sheet” ({τ>0}) of the two-sheeted hyperboloid in Rn+1 defined in coordinates (ξ1, ··· ,ξn,τ) by the equation 2 2 2 1 ∗ n+1 n+1 τ −|ξ| = R , with the metric hR = ı m, where ı : HR ,→ R is inclusion, and m is the Minkowski metric on Rn+1. n n (2) Poincar´eModel BR is the ball of radius R in R , with the metric given in coordinates (u1, ··· ,un)by (du1)2 + ···+(dun)2 h2 =4R4 . R (R2 −|u|2)2 n n (3) Poincar e Half-space Model UR is the upper half-space in R defined in coordinates (x1, ··· ,xn,y)by{y>0}, with the metric (dx1)2 + ···+(dxn−1)2 + dy2 h3 = R2 . R y2 To construct an isometry between the two metrics given in (2) and (3), we first construct an explicit diffeomorphism n n κ : BR → UR. Writing the coordinates on the ball as (u1, ··· ,un−1,v)=(u, v), κ can be written as 2R2u R2 −|u|2 − v2 κ(u, v)=(x, y)= ,R . |u|2 +(v − R)2 |u|2 +(v − R)2 It is straightforward to check that its inverse is 2R2x |x|2 + |y|2 − R2 κ−1(x, y)=(u, v)= ,R . |x|2 +(y + R)2 |x|2 +(y + R)2 Hence κ is a diffeomorphism, called the generalized Cayley transform. ∗ 3 2 Sow that κ hR = hR. 4 Problem 8. Let M(n, R) denote the vector space of n × n matrices with real entries. For a matrix A ∈ M(n, R), let AT be its transpose and let O(n)={A ∈ GL(n, C),AT A =1}, the orthogonal group. SL(n)={A ∈ GL(n, C),AT A =1}, the special linear group. SO(n)={A ∈ O(n) : det A =1}, the special oethogonal group. (1) Let S(n, R) denote the set of symmetric n × n matrices. Define Φ : GL(n, R) → S(n, R)by Φ(A)=AT A. Let A ∈ O(n) be arbitrary. To compute the pushforward Φ∗ : TAGL(n, R) → TΦ(A)S(n, R), we choose arbitrarily n × n matrix B and consider the curve γ(t)=A + tB. 0 0 T T (i) Show that Φ∗B =Φ∗γ (0) = (Φ ◦ γ) (0) = B A + A B. (ii) Show that the identity matrix In is a regular value of Φ; i.e. show that Φ∗ : TAGL(n, R) → TΦ(A)S(n, R) is surjective for each A ∈ O(n). −1 (iii) Show that O(n)=Φ (In) is an embedded submanifold of GL(n, R). (2) Let det : GL(n, R) → R \{0} denote the determinant function. i (i) Using the matrix entries (Xj ) as global coordinates on GL(n, R), show that ∂ −1 i i (det X) = (det X)(X )j . ∂Xj (Hint: Expand det X by minors along the ith column and use Cramer’s rule.) (ii) Show that the differential of the determinant function is −1 d(det)X (B) = (det X)tr(X B) ∼ i for X ∈ GL(n, R) and B ∈ TX GL(n, R) = M(n, R), where trX = Pi Xi is the trace of X. (iii) Show that det : GL(n, R) → R \{0} is a submersion. (Hint: d(det)AA =6 0 for each A ∈ GL(n, R).) (iv) Show that SL(n, R) is an embedded submanifold of GL(n, R). (v) Show that Lie(SL(n, R)) =∼ {A ∈ M(n, R), trA =0}. (3) (i) Show that SO(n, R) is an embedded Lie submanifold of GL(n, R). (ii) Show that Lie(SO(n)) =∼ Lie(O(n)) ∩ Lie(SL)(n, R). 5 Problem 9. Let M(n, C) be the vector space of all the n×n matrices of complex entries. Let GL(n, C) is the Lie group of all complex n×n matrices of nonvanishing determinant. Let A∗ = AT be the conjugate transpose of A and U(n)={A ∈ GL(n, C),A∗A =1}, the unitary group, SU(n)={A ∈ U(n) : det A =1}, the special unitary group, SL(n, C)={A ∈ GL(n, C) : det A =1}, the complex special linear group. (1) Show that Lie(U(n)) =∼{A ∈ M(n, C): A∗ + A =0}, Lie(SL(n, C)) =∼{A ∈ M(n, C), trA =0}, Lie(SU(n)) =∼Lie(U(n)) ∩ Lie(SL)(n, C). (2) (i) Show that the following matrices are a basis for Te of Lie(SU(2)): 1 0 i 1 0 −1 1 i 0 J = ,J= ,J= .